Solving linear equations efficiently is one of the important and well studied problems in the field of linear algebra, and there are several elegant approaches in the literature to address this problem. The Krylov subspace methods, a family of iterative methods for solving linear equations, are most popular in the literature for solving \(A\mathbf {z} = \mathbf {u}\)  efficiently just by using the matrix-vector product \(A\mathbf {u}\)  (or, \(A^{^{\hspace{-0.83328pt}\mathsf {T}}}\mathbf {u}\) ), see for instance, [1]}. Among these methods, the conjugate gradient method is well-known to work efficiently for the problems where \(A\)  is symmetric positive-definite and can achieve a close approximation to \(A^{-1}\mathbf {u}\)  within \(O(p^2)\)  time. Since, \(L_{\mathbf {t}}\)  is a symmetric positive-definite matrix, we can simply use any standard software package for the conjugate gradient method in the implementation of COMBSS.
